The amount of waste a company produces, W, in tons per week, is approximated by W =

In Problems 14, explain in words what the integral represents and give units. =3 1 v(t) dt, where v(t) is velocity in meters/sec and t is time in seconds.

In Problems 14, explain in words what the integral represents and give units. =6 0 a(t) dt, where a(t) is acceleration in km/hr2 and t is time in hours.

In Problems 14, explain in words what the integral represents and give units. =2011 2005 f(t) dt, where f(t) is the rate at which world population is growing in year t, in billion people per year.

In Problems 14, explain in words what the integral represents and give units. =5 0 s(x) dx, where s(x) is rate of change of salinity (salt concentration) in gm/liter per cm in sea water, and where x is depth below the surface of the water in cm.

Oil leaks out of a tanker at a rate of r = f(t) gallons per minute, where t is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Pollution is removed from a lake on day t at a rate of f(t) kg/day. (a) Explain the meaning of the statement f(12) = 500. (b) If =15 5 f(t) dt = 4000, give the units of the 5, the 15, and the 4000. (c) Give the meaning of =15 5 f(t) dt = 4000.

Annual coal production in the US (in billion tons per year) is given in the table.6 Estimate the total amount of coal produced in the US between 1997 and 2009. If r = f(t) is the rate of coal production t years since 1997, write an integral to represent the 19972009 coal production.

The following table gives the US emissions, H(t), of hydrofluorocarbons, or super greenhouse gasses, in teragrams equivalent of carbon dioxide, with t in years since 2000.7 (a) What are the units and meaning of =10 0 H(t)dt? (b) Estimate =10 0 H(t)dt. Year 2000 2002 2004 2006 2008 2010 H(t) 7104 7022 7163 7159 7048 6822

World annual natural gas8 consumption, N, in millions of metric tons of oil equivalent, is approximated by N = 1770 + 53t, where t is in years since 1990. (a) How much natural gas was consumed in 1990? In 2010? (b) Estimate the total amount of natural gas consumed during the 20-year period from 1990 to 2010.

Solar photovoltaic (PV) cells are the worlds fastestgrowing energy source. In year t since 2007, PV cells were manufactured worldwide at a rate of S = 3.7e0.61t gigawatts per year.9 Estimate the total solar energygenerating capacity of the PV cells manufactured between 2007 and 2011.

Problems 1114 show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Compute the change in position between times t = 0 and t = 5 seconds.

Problems 1114 show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Compute the change in position between times t = 0 and t = 5 seconds.

Problems 1114 show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Compute the change in position between times t = 0 and t = 5 seconds.

Problems 1114 show the velocity, in cm/sec, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Compute the change in position between times t = 0 and t = 5 seconds.

Your velocity is v(t) = ln(t2+1) ft/sec for t in seconds, 0 t 3. Find the distance traveled during this time.

Figure 5.48 shows the length growth rate of a human fetus. (a) What feature of a graph of length as a function of age corresponds to the maximum in Figure 5.48? (b) Estimate the length of a baby born in week 40. 8 16 24 32 40 0 1 2 age of fetus (weeks after last menstruation) length growth rate (cm/week) Figure 5.48

A forest fire covers 2000 acres at time t = 0. The fire is growing at a rate of 8t acres per hour, where t is in hours. How many acres are covered 24 hours later?

Water is pumped out of a holding tank at a rate of 5 5e0.12t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later?

With t in seconds, the velocity of an object is v(t) = 10 + 8t t2 m/sec. (a) Represent the distance traveled during the first 5 seconds as a definite integral and as an area. (b) Estimate the distance traveled by the object during the first 5 seconds by estimating the area. (c) Calculate the distance traveled.

A bungee jumper leaps off the starting platform at time t = 0 and rebounds once during the first 5 seconds. With velocity measured downward, for t in seconds and 0 t 5, the jumpers velocity is approximated10 by v(t) = 4t2 +16t meters/sec. (a) How many meters does the jumper travel during the first five seconds? (b) Where is the jumper relative to the starting position at the end of the five seconds? (c) What does =5 0 v(t) dt represent in terms of the jump?

After a foreign substance is introduced into the blood, the rate at which antibodies are made is given by r(t) = t t2 + 1 thousands of antibodies per minute, where time, t, is in minutes. Assuming there are no antibodies present at time t = 0, find the total quantity of antibodies in the blood at the end of 4 minutes.

Figure 5.49 gives your velocity during a trip starting from home. Positive velocities take you away from home and negative velocities take you toward home.Where are you at the end of the 5 hours? When are you farthest from home? How far away are you at that time? 1 2 3 4 5 20 10 10 20 30 40 t (hours) v (km/hr) Figure 5.49

A bicyclist pedals along a straight road with velocity, v, given in Figure 5.50. She starts 5 miles from a lake; positive velocities take her away from the lake and negative velocities take her toward the lake. When is the cyclist farthest from the lake, and how far away is she then? 1 10 0 10 20 30 t (hours) v (mph) Figure 5.50

Figure 5.51 shows the rate of growth of two trees. If the two trees are the same height at time t = 0, which tree is taller after 5 years? After 10 years?

(a) Write each of the following areas in Figure 5.52 from 2000 to 2015 as an integral and explain its significance for the fund. (i) Under the income curve (ii) Under the expenditure curve (iii) Between the income and expenditure curves (b) Use Figure 5.53 to estimate the area between the two curves from 2000 to 2015.

Decide when the value of the fund is projected to be a maximum using (a) Figure5.52 (b) Figure5.53

Express the projected increase in value of the fund from 2000 to 2030 as an integral.

The rates of consumption of stores of protein and fat in the human body during 8 weeks of starvation are shown in Figure 5.54. Does the body burn more fat or more protein during this period?

Figure 5.55 shows the number of sales per month made by two salespeople.Which person has the most total sales after 6 months? After the first year? At approximately what times (if any) have they sold roughly equal total amounts? Approximately how many total sales has each person made at the end of the first year? 2 4 6 8 10 12 10 20 30 40 50 t (months) number of sales per month Salesperson B Salesperson A Figure 5.55

The birth rate, B, in births per hour, of a bacteria population is given in Figure 5.56. The curve marked D gives the death rate, in deaths per hour, of the same population. (a) Explain what the shape of each of these graphs tells you about the population. (b) Use the graphs to find the time at which the net rate of increase of the population is at a maximum. (c) At time t = 0 the population has size N. Sketch the graph of the total number born by time t. Also sketch the graph of the number alive at time t. Estimate the time at which the population is a maximum. 5 10 15 20 time (hours) bacteria/hour B D Figure 5.56

Height velocity graphs are used by endocrinologists to follow the progress of children with growth deficiencies. Figure 5.57 shows the height velocity curves of an average boy and an average girl between ages 3 and 18. (a) Which curve is for girls and which is for boys? Explain how you can tell. (b) About how much does the average boy grow between ages 3 and 10? (c) The growth spurt associated with adolescence and the onset of puberty occurs between ages 12 and 15 for the average boy and between ages 10 and 12.5 for the average girl. Estimate the height gained by each average child during this growth spurt. (d) When fully grown, about how much taller is the average man than the average woman? (The average boy and girl are about the same height at age 3.)

(a) If the body is bled 2 liters, how much blood is pumped during the three hours leading to death? (b) If f(t) is the pumping rate in liters per minute at time t hours, express your answer to part (a) as a definite integral. (c) How much more blood would have been pumped during the same time period if there had been no bleeding? Illustrate your answer on the graph.

(a) If the body is bled 1 liter, how much blood is pumped during the three hours leading to full recovery? (b) If g(t) is the pumping rate in liters per minute at time t hours, express your answer to part (a) as a definite integral. (c) How much more blood would have been pumped during the same time period if there had been no bleeding? Show your answer as an area on the graph.

The amount of waste a company produces, W, in tons per week, is approximated by W = 3.75e0.008t, where t is in weeks since January 1, 2005. Waste removal for the company costs $15/ton. How much did the company pay for waste removal during the year 2005?

Figure 5.59 shows plasma concentration curves for two drugs used to slow a rapid heart rate. Compare the two products in terms of level of peak concentration, time until peak concentration, and overall bioavailability. Product B Product A hours concentration of drug in plasma Figure 5.59

Figure 5.60 compares the concentration in blood plasma for two pain relievers. Compare the two products in terms of level of peak concentration, time until peak concentration, and overall bioavailability. Product A Product B hours concentration of drug in plasma Figure 5.60

Draw plasma concentration curves for two drugs A and B if product A has the highest peak concentration, but product B is absorbed more quickly and has greater overall bioavailability.

A two-day environmental cleanup started at 9 am on the first day. The number of workers fluctuated as shown in Figure 5.61. If the workers were paid $10 per hour, how much was the total personnel cost of the cleanup? 8 16 24 32 40 48 10 20 30 40 50 hours workers Figure 5.61

Suppose in Problem 38 that the workers were paid $10 per hour for work during the time period 9 am to 5 pm and were paid $15 per hour for work during the rest of the day. What would the total personnel costs of the cleanup have been under these conditions?

At the site of a spill of radioactive iodine, radiation levels were four times the maximum acceptable limit, so an evacuation was ordered. If R0 is the initial radiation level (at t = 0) and t is the time in hours, the radiation level R(t), inmillirems/hour, is given by R(t) = R0(0.996)t . (a) How long does it take for the site to reach the acceptable level of radiation of 0.6 millirems/hour? (b) How much total radiation (in millirems) has been emitted by that time?

If you jump out of an airplane and your parachute fails to open, your downward velocity (in meters per second) t seconds after the jump is approximated by v(t) = 49(1 (0.8187)t). (a) Write an expression for the distance you fall in T seconds. (b) If you jump from 5000 meters above the ground, estimate, using trial and error, how many seconds you fall before hitting the ground.

The Montgolfier brothers (Joseph and Etienne) were eighteenth-century pioneers of hot-air ballooning. Had they had the appropriate instruments, they might have left us a record, like that shown in Figure 5.62, of one of their early experiments. The graph shows their vertical velocity, v, with upward as positive. (a) Over what intervals was the acceleration positive? Negative? (b) What was the greatest altitude achieved, and at what time? (c) This particular flight ended on top of a hill. How do you know that it did, and what was the height of the hill above the starting point?